On some arithmetic properties of Siegel functions
type: Item
instance of: scholarly article
title: On some arithmetic properties of Siegel functions
zbMATH Open document ID: 1228.11057
author: Ja Kyung Koo
author: Dong Hwa Shin
published in: Mathematische Zeitschrift
publication date: 2010-02-15T00:00:00Z
review text: For \(r=(r_1,r_2)\in\mathbb Q^2\backslash\mathbb Z^2\), let the Siegel function \(g_r\) be defined by \[ g_r(\tau)=-q_\tau^{B_2(r_1)/2}e^{\pi ir_2(r_1-1)}(1-q_z)\prod_{n=1}^\infty (1-q_\tau^nq_z)(1-q_\tau^nq_z^{-1}), \] where \(q_z=e^{2\pi iz}\) with \(z=r_1\tau+r_2\) and \(B_2(x)=x^2-x+1/6\). Then it is known that a suitable product of Siegel functions will be a modular function on \(\Gamma(N)\). (See [\textit{D. S. Kubert} and \textit{S. Lang}, Modular units. New York etc.: Springer-Verlag (1981; Zbl 0492.12002)] for the necessary and sufficient conditions.) In the paper under review, the authors study several problems related to Siegel functions. Firstly, they show that the field of modular functions on \(\Gamma(N)\) is generated by the elliptic \(j\)-function, \(g^{12N}_{(1/N,0)}\), and \(g^{12N}_{(0,1/N)}\). They also discuss the cases of \(\Gamma_1(N)\) and \(\Gamma_0(N)\). Secondly, the authors determine when a product of Siegel functions is integral over \(\mathbb Z[j]\). As an application, they show that the reciprocals of the values of Ramanujan's cubic continued fraction at imaginary quadratic numbers are all algebraic integers. Finally, the authors use special values of Siegel functions to explicitly construct ray class fields of imaginary quadratic number fields.
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11G16
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5671533
zbMATH Keywords: modular units
zbMATH Keywords: Siegel functions
zbMATH Keywords: class fields
reviewed by: Yifan Yang
MaRDI profile type: Publication
full work available at URL: s00209-008-0456-9
OpenAlex ID: W2799994795
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DOI: 10.1007/S00209-008-0456-9
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description: scientific article; zbMATH DE number 5671533